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Unformatted text preview: MMAE 350 D. Rempfer Problem Set VII Page 1 of 2 Due Date: November 20, 2006 1. We want to investigate the dynamics of the simple swinging pendulum (see picture), which is thought of as a point mass that is suspended on a rod of zero mass and length l , in a gravitational field with an acceleration of g . The angle between the vertical coordinate and the pendulum is denoted by Θ . a. Use Newton’s second law to derive the di ff erential equation that de- scribes the change of the angle Θ with time. b. Linearize the nonlinear di ff erential equation you have obtained above. (Hint: Use a Taylor-series expansion for the function on the right-hand side of your di ff erential equation.) Q m l c. What is the exact solution of the linearized di ff erential equation, for Θ ( t = 0) = 4 π/ 5, d Θ /dt ( t = 0) = 0? d. Using Heun’s predictor-corrector method, solve the exact di ff erential equation from part a. of this problem numerically, for the initial condition Θ ( t = 0) = 4 π/ 5, d Θ /dt ( t = 0) = 0, using g/ (2 π` ) = 1. Try to integrate over a time of t e = 4 π , using step sizes h 1...
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- Spring '05
- Chaos Theory, step size, fourth-order Runge-Kutta method, D. Rempfer